Kagome点阵夹芯板的分岔和混沌动力学研究

As a new type of sandwich material, the truss core sandwich structure is widely used in aerospace and other high technology fields due to its excellent comprehensive performance. It has the important theory and practical significance to investigate this structure. In this project, the theories of bifurcation and chaos of high-dimensional nonlinear systems are applied to the studies of the sandwich plate with Kagome truss core. Including: we will investigate the stability and local bifurcations for the Kagome truss core sandwich plate and try to obtain the stability regions of the initial equilibrium solution and the critical bifurcation curves in the case of 1:3 internal resonance. The existence of Shilnikov-type single-pulse homoclinic orbits and multi-pulse homoclinic orbits are investigated by using the global perturbation method and extended Melnikov method with the case of 1:1 internal resonance. We obtain the explicit sufficient conditions for the existence of the Shilnikov-type multi-pulse homoclinic orbits by applying the energy-phase method in the case of 1:2 internal resonance. Some criteria are established for the Smale horesshoes chaos in the system. The result will reveal the possible mechanism for the existence of bifurcations and complex chaotic motions in Kagome truss core sandwich plate rigorously and analytically. It provides the corresponding theoretical basis for the analysis and parameter design of the model.

点阵夹芯结构作为一种新型夹芯材料，因其优异的综合性能，被广泛应用在航空航天等高科技领域。对这类结构模型非线性动力学性能的研究有重要的理论和实际意义。本项目拟结合一类Kagome轻质点阵夹芯板模型，用高维非线性系统动力学理论研究其分岔和混沌行为。具体包括：在1:3内共振时，研究系统的稳定性和局部分岔行为，讨论退化平衡点的稳定区域和分岔曲线；在1:1内共振时，利用全局摄动法和广义Melnikov方法讨论夹芯板存在Shilnikov型单脉冲和多脉冲同宿轨的若干参数条件；在1:2内共振时，利用能量相位法研究点阵夹芯板在有阻尼时Shilnikov型多脉冲同宿轨的存在性，建立系统发生Smale马蹄混沌的若干判据。研究结果将严格解析的揭示Kagome点阵夹芯板发生分岔和混沌运动的复杂性机制，为模型的分析和参数设计提供相应的理论基础。

夹芯板和圆柱壳等轻质结构材料广泛应用在许多工程及高科技领域，对其非线性动力学行为的研究具有重要的理论意义和应用价值。本课题主要应用高维非线性动力学理论研究Kagome点阵夹芯板、圆柱壳、弹性梁等模型的分岔和混沌动力学行为。研究工作主要集中在以下几个方面：.(1) 研究了一类受横向载荷和面内载荷联合作用下Kagome点阵夹芯板的混沌动力学行为，利用改进的Melnikov方法，得到非自治系统存在多脉冲混沌运动的参数条件；.(2) 研究了带压电制动器的圆柱壳在1:2内共振下的全局分岔和混沌动力学行为，利用能量相位法，获得耗散系统存在Shilnikov型多脉冲同宿轨的充分条件，分析了阻尼、激励参数对模态之间能量传递的影响，建立了系统发生Smale马蹄混沌的若干判据；.(3) 研究了一类受横向激励和轴向激励的弹性梁的次谐分岔和混沌行为，获得了弹性梁模型发生次谐分岔和超次谐分岔的参数条件，给出了系统混沌区域和非混沌区域的分界曲线。.(4) 研究了复合材料圆柱壳在1:1内共振下的稳定性和局部分岔问题，得到系统发生静态分岔、Hopf分岔及2D胎面的参数条件及分岔曲线。.研究结果展示了几类非线性系统丰富的动力学现象，也为模型的分析和参数设计提供了相应的理论指导。